Theorem sbcom2 992

Description: Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint).

Assertion

Ref	Expression
sbcom2	|- ([w / z][y / x]ph <-> [y / x][w / z]ph)

Distinct variable group(s):   x,z   x,w   y,z

Proof of Theorem sbcom2

Step	Hyp	Ref	Expression
1		alcom 715	. . . . . 6 |- (A.zA.x(x = y -> (z = w -> ph)) <-> A.xA.z(x = y -> (z = w -> ph)))
2		bi2.04 141	. . . . . . . . 9 |- ((x = y -> (z = w -> ph)) <-> (z = w -> (x = y -> ph)))
3	2	bial 695	. . . . . . . 8 |- (A.x(x = y -> (z = w -> ph)) <-> A.x(z = w -> (x = y -> ph)))
4		19.21v 942	. . . . . . . 8 |- (A.x(z = w -> (x = y -> ph)) <-> (z = w -> A.x(x = y -> ph)))
5	3, 4	bitr 151	. . . . . . 7 |- (A.x(x = y -> (z = w -> ph)) <-> (z = w -> A.x(x = y -> ph)))
6	5	bial 695	. . . . . 6 |- (A.zA.x(x = y -> (z = w -> ph)) <-> A.z(z = w -> A.x(x = y -> ph)))
7		19.21v 942	. . . . . . 7 |- (A.z(x = y -> (z = w -> ph)) <-> (x = y -> A.z(z = w -> ph)))
8	7	bial 695	. . . . . 6 |- (A.xA.z(x = y -> (z = w -> ph)) <-> A.x(x = y -> A.z(z = w -> ph)))
9	1, 6, 8	3bitr3 156	. . . . 5 |- (A.z(z = w -> A.x(x = y -> ph)) <-> A.x(x = y -> A.z(z = w -> ph)))
10	9	a1i 7	. . . 4 |- ((-. A.x x = y /\ -. A.z z = w) -> (A.z(z = w -> A.x(x = y -> ph)) <-> A.x(x = y -> A.z(z = w -> ph))))
11		sb4b 862	. . . . 5 |- (-. A.z z = w -> ([w / z][y / x]ph <-> A.z(z = w -> [y / x]ph)))
12		sb4b 862	. . . . . . 7 |- (-. A.x x = y -> ([y / x]ph <-> A.x(x = y -> ph)))
13	12	imbi2d 464	. . . . . 6 |- (-. A.x x = y -> ((z = w -> [y / x]ph) <-> (z = w -> A.x(x = y -> ph))))
14	13	bialdv 935	. . . . 5 |- (-. A.x x = y -> (A.z(z = w -> [y / x]ph) <-> A.z(z = w -> A.x(x = y -> ph))))
15	11, 14	sylan9bbr 419	. . . 4 |- ((-. A.x x = y /\ -. A.z z = w) -> ([w / z][y / x]ph <-> A.z(z = w -> A.x(x = y -> ph))))
16		sb4b 862	. . . . 5 |- (-. A.x x = y -> ([y / x][w / z]ph <-> A.x(x = y -> [w / z]ph)))
17		sb4b 862	. . . . . . 7 |- (-. A.z z = w -> ([w / z]ph <-> A.z(z = w -> ph)))
18	17	imbi2d 464	. . . . . 6 |- (-. A.z z = w -> ((x = y -> [w / z]ph) <-> (x = y -> A.z(z = w -> ph))))
19	18	bialdv 935	. . . . 5 |- (-. A.z z = w -> (A.x(x = y -> [w / z]ph) <-> A.x(x = y -> A.z(z = w -> ph))))
20	16, 19	sylan9bb 418	. . . 4 |- ((-. A.x x = y /\ -. A.z z = w) -> ([y / x][w / z]ph <-> A.x(x = y -> A.z(z = w -> ph))))
21	10, 15, 20	3bitr4d 424	. . 3 |- ((-. A.x x = y /\ -. A.z z = w) -> ([w / z][y / x]ph <-> [y / x][w / z]ph))
22	21	exp 291	. 2 |- (-. A.x x = y -> (-. A.z z = w -> ([w / z][y / x]ph <-> [y / x][w / z]ph)))
23		eq5 824	. . . 4 |- (A.x x = y -> A.zA.x x = y)
24		sbequ12 865	. . . . 5 |- (x = y -> (ph <-> [y / x]ph))
25	24	a4s 682	. . . 4 |- (A.x x = y -> (ph <-> [y / x]ph))
26	23, 25	bisbd 897	. . 3 |- (A.x x = y -> ([w / z]ph <-> [w / z][y / x]ph))
27		sbequ12 865	. . . 4 |- (x = y -> ([w / z]ph <-> [y / x][w / z]ph))
28	27	a4s 682	. . 3 |- (A.x x = y -> ([w / z]ph <-> [y / x][w / z]ph))
29	26, 28	bitr3d 408	. 2 |- (A.x x = y -> ([w / z][y / x]ph <-> [y / x][w / z]ph))
30		sbequ12 865	. . . 4 |- (z = w -> ([y / x]ph <-> [w / z][y / x]ph))
31	30	a4s 682	. . 3 |- (A.z z = w -> ([y / x]ph <-> [w / z][y / x]ph))
32		eq5 824	. . . 4 |- (A.z z = w -> A.xA.z z = w)
33		sbequ12 865	. . . . 5 |- (z = w -> (ph <-> [w / z]ph))
34	33	a4s 682	. . . 4 |- (A.z z = w -> (ph <-> [w / z]ph))
35	32, 34	bisbd 897	. . 3 |- (A.z z = w -> ([y / x]ph <-> [y / x][w / z]ph))
36	31, 35	bitr3d 408	. 2 |- (A.z z = w -> ([w / z][y / x]ph <-> [y / x][w / z]ph))
37	22, 29, 36	pm2.61ii 113	1 |- ([w / z][y / x]ph <-> [y / x][w / z]ph)

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672   = weq 797  [wsb 852

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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